We describe inertial endomorphisms of an abelian group A, that is endomorphisms\ud ϕ with the property |(ϕ(X) + X)/X| < ∞ for each X ≤ A. They form a ring\ud I E(A) containing the ideal F(A) formed by the so-called finitary endomorphisms, the ring\ud of power endomorphisms and also other non-trivial instances.We show that the quotient ring\ud I E(A)/F(A) is commutative. Moreover, inertial invertible endomorphisms form a group,\ud provided A has finite torsion-free rank. In any case, the group I Aut(A) they generate is\ud commutative modulo the group FAut (A) of finitary automorphisms, which is known to be\ud locally finite. We deduce that I Aut(A) is locally-(center-by-finite). Also, we consider the\ud lattice dual property, that is |X/(X ∩ϕ(X))| < ∞for each X ≤ A and show that this implies\ud the above one, provided A has finite torsion-free rank

We study the group IAut(A) generated by the inertial automorphisms of an abelian group A, that is, automorphisms γ with the property that each subgroup H of A has finite index in the subgroup generated by H and Hγ. Clearly, IAut(A) contains the group FAut(A) of finitary automorphisms of A, which is known to be locally finite. In a previous paper, we showed that IAut(A) is (locally finite)-by-abelian. In this paper, we show that IAut(A) is also metabelian-by-(locally finite). In particular, IAut(A) has a normal subgroup Γ such that IAut(A)/Γ is locally finite and Γ ′ is an abelian periodic subgroup whose all subgroups are normal in Γ. In the case when A is periodic, IAut(A) results to be abelian-by-(locally finite) indeed, while in the general case it is not even (locally nilpotent)-by-(locally finite). Moreover, we provide further details about the structure of IAut(A) in some other cases for A.If A is any periodic abelian group, then PAut(A) is the cartesian product of all theAccording to [10], an automorphism γ is called an (invertible) multiplication of A if and only if it is a power automorphism of A, if A is periodic, or -when A is non-periodicthere exist coprime integers m, n such that (na)γ = ma, for each a ∈ A. In the latter case, we have mnA = A and A π(mn) = 0 and -with an abuse of notation-we will write γ = m/n. We warn that we are using the word "multiplication" in a way different from [14]. Invertible multiplications of A form a subgroup which is a central subgroup of Aut(A).

If H is a subgroup of an abelian group G and φ ∈ End(G), H is called φ-inert (and φ is H-inertial) if φ(H) ∩ H has finite index in the image φ(H). The notion of φ-inert subgroup arose and was investigated in a relevant way in the study of the so called intrinsic entropy of an endomorphism φ, while inertial endo-morphisms (these are endomorphisms that are H-inertial for every subgroup H) were intensively studied by Rinauro and the first named author.A subgroup H of an abelian group G is said to be fully inert if it is φ-inert for every φ ∈ End(G). This property, inspired by the “dual” notion of inertial endomorphism, has been deeply investigated for many different types of groups G. It has been proved that in some cases all fully inert subgroups of an abelian group G are commensurable with a fully invariant subgroup of G (e.g., when G is free or a direct sum of cyclic p-groups). One can strengthen the notion of fully inert subgroup by defining H to be uniformly fully inert if there exists a positive integer n such that |(H + φH)/H| ≤ n for every φ ∈ End(G). The aim of this paper is to study the uniformly fully inert subgroups of abelian groups. A natural question arising in this investigation is whether such a subgroup is commensurable with a fully invariant subgroup. This paper provides a positive answer to this question for groups belonging to several classes of abelian groups.

An endomorphisms $\p$ of an abelian group $A$ is said inertial if each subgroup $H$ of $A$ has finite index in $H+\varphi (H)$. We\ud study the ring of inertial endomorphisms of an abelian group. We obtain a satisfactory description modulo\ud the ideal of finitary endomorphisms.\ud Also the corresponding problem for vector spaces is considered

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